The geometric mean (GM) is the common cost or mean which indicates the valuable tendency of the set of numbers with the aid of using locating the product of their values.

In arithmetic and statistics, a measure of valuable inclinations describes the precis of complete statistics set value. The maximum critical measure of valuable inclination is mean, median, mode, and range. Among these, the mean of the statistics set gives the general concept of the statistics.

The mean defines the common numbers in the data set. The one-of-a-kind sorts of mean are Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). In this article, we will discuss the definition, formula, of geometric mean with solved examples.

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**Geometric Mean**

**Geometric Mean**

The geometric mean (GM) is the common cost or mean which indicates the valuable tendency of the set of numbers by taking the roots of the product in their values.

Generally, we simply multiply all the given numbers and take the __n__^{th}__ root__ of the given number, where n is the total number of values. It can also be described as the n^{th} root of the product of all the values. Keep in mind that geometric mean is different from arithmetic means.

A geometric mean is used when working with percentage while on the other hand arithmetic means work directly with the values themselves. In simple words, in geometric means, we multiply all the numbers in series and then take the nth root of the series of numbers. On the other hand, in arithmetic mean, we simply sum the series and then divide that sum with the total numbers involved in that series.

**Geometric Mean Formula**

**Geometric Mean Formula**

To calculate the geometric mean, we use a formula known as the geometric mean formula. Let a_{1}, a_{2}, a_{3}, …, a_{n} is a series of numbers, we want to calculate the geometric mean then we use a formula in which we multiply all the numbers and take n^{th}_{ }root.

**GM = (a**_{1}** * a**_{2}** * a**_{3}** * … * a**_{n}**)**^{1/n}

This formula can be termed in logarithm as,

log GM = log (a_{1} * a_{2} * a_{3} * … * a_{n})^{1/n}

log GM = 1/n log (a_{1} * a_{2} * a_{3} * … * a_{n})

log GM = 1/n (log a_{1} + log a_{2} + log a_{3} + … + log a_{n})

log GM = Σ log a_{i} /n

Taking antilog both sides.

**GM = Antilog ****Σ**** log a**_{i}** /n**

As log and antilog should be canceled.

**How to calculate Geometric Mean?**

**How to calculate Geometric Mean?**

To learn how to calculate geometric mean using formula let us take some examples to understand more accurately.

**Example 1**

Calculate the geometric mean of 2, 4, 6, 8.

**Solution**** **

** Step 1:** write the formula of the Geometric Mean.

**GM = (a**_{1}** * a**_{2}** * a**_{3}** * … * a**_{n}**)**^{1/n}

** Step 2:** Identify the values.

2, 4, 6, 8

n = 4

** Step 3:** Put the values in the formula.

GM = (2 * 4 * 6 * 8)^{1/4}

= (384)^{1/4}

= (384)^{0.25}

Evaluate the exponent or use __exponent calculator__ to solve numbers with exponents.

= 4.43

**Example 2**

Calculate the geometric mean of 12, 14, 16, 18.

**Solution**** **

** Step 1:** write the formula of the Geometric Mean.

**GM = (a**_{1}** * a**_{2}** * a**_{3}** * … * a**_{n}**)**^{1/n}

** Step 2:** Identify the values.

12, 14, 16, 18

n = 4

** Step 3:** Put the values in formula.

GM = (12 * 14 * 16 * 18)^{1/4}

= (384)^{1/4}

= (48384)^{0.25}

= 14.83

__Geometric Mean Calculator__ is an online tool that will assist you in calculating geometric mean and gives you a step-by-step solution at one click.

**Example 3**

Calculate the geometric mean of 5, 10, 15, 20, 25, 30, 35.

**Solution**** **

** Step 1:** write the formula of Geometric Mean.

**GM = (a**_{1}** * a**_{2}** * a**_{3}** * … * a**_{n}**)**^{1/n}

** Step 2:** Identify the values.

5, 10, 15, 20, 25, 30, 35

n = 7

** Step 3:** Put the values in formula.

GM = (5 * 10 * 15 * 20 * 25 * 30 * 35)^{1/4}

= (393750000)^{1/7}

= (393750000)^{0.1428}

= 16.90

**Example 4**

Calculate the geometric mean of 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37

**Solution**** **

** Step 1:** Write the formula of Geometric Mean.

**GM = (a**_{1}** * a**_{2}** * a**_{3}** * … * a**_{n}**)**^{1/n}

** Step 2:** Identify the values.

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37

n = 7

** Step 3:** Put the values in formula.

GM = (2 * 3 * 5 * 7 * 9 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37)^{1/13}

= (6,67866432e13)^{1/7}

= (6.67866432e13)^{0.08}

= 11.57

**Applications of Geometric Mean**

**Applications of Geometric Mean**

Advantages of geometric mean have more than the arithmetic mean. The geometric mean is used in many fields of mathematics, statistics, and science.

Applications of the geometric mean are given below.

- It is utilized in stock indexes due to the fact a number of value line indexes that are utilized by the economic department make use of geometric mean.
- To find the annual return on the investment portfolio.
- The geometric mean is used in finance to calculate the mean growth rates.
- The geometric mean is also used in biological studies to calculate the bacterial growth rate or to calculate cell division.

**Tips and tricks to find Geometric Mean**

**Tips and tricks to find Geometric Mean**

Geometric mean has some tips and tricks let us discuss them.

- The geometric mean of the given number of series is always smaller than the arithmetic means of that given series.
- If we have two series, then the product of corresponding numbers of the geometric mean is equal to the product of their geometric mean.
- If each value in the given series is interchanged then the product of values remains the same even by changing the numbers.
- If we have two series, then the ratio of corresponding numbers of the geometric mean is equal to the ratio of their geometric mean.

**Summary **

**Summary**

The geometric mean is a type of mean. We can calculate problems related to geometric mean easily by using the formula of the geometric mean. In this article, we discussed the definition, formula, applications, and tips of geometric series.